You can use Audacity. Select the "Windows WASAPI" in the recording interface, and select "speaker blah blah blah (loopback)" for recording. In this way, Audacity will record whatever audio flow through your speaker instead of microphone.

Then Start+run mmsys.cpl -> recording -> microphone -> properties -> listen -> enable "listen to this device". This will force the speaker to play whatever sound recorded by your microphone, and in this way, your own voice will be played through speaker and hence captured by audacity due to the "Windows WASAPI" and the "loopback".

The only drawback is that you have to hear your own voice in the speaker when you talk through Skype. But since "Stereo Mix" is no longer available in many modern computers running latest Windows with integrated sound cards, this is perhaps the best and easiest way without installing any other third party craps.